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On the factorization theorem for the tensor product of integrable distributions

  • Oswaldo Flores-MedinaEmail author
  • Juan H. Arredondo
  • Juan A. Escamilla-Reyna
  • Francisco J. Mendoza-Torres
Original Paper

Abstract

In this paper, we applied the Factorization Theorem of W. Rudin and H. Cohen for obtaining a factorization of the tensor product of the space \(\overline{ HK ( {\mathbb {R}} ) }\) with itself, where \(\overline{ HK ( {\mathbb {R}} ) }\) is the completion of the space of the Henstock-Kurzweil integrable functions. This generalizes the results over the factorizations of \(L^1 ({\mathbb {R}})\) and \(\overline{ HK ( {\mathbb {R}} ) }\). In particular, we prove that for the Banach algebra \( \overline{ \big ( HK({\mathbb {R}})\cap BV({\mathbb {R}}) \big ) \otimes _\gamma \big ( HK({\mathbb {R}})\cap BV({\mathbb {R}}) \big ) } \) contained in \( \overline{ HK({\mathbb {R}}) \otimes _\gamma HK({\mathbb {R}}) }\), the Factorization Theorem does not hold. Similar results are therefore valid for the space \(\overline{ {\mathcal {A}}_C ({\mathbb {R}}) \otimes _\gamma {\mathcal {A}}_C ({\mathbb {R}}) }\). Moreover, we build a new integral which is applied to extend some properties of the Fourier Transform on the classic space \(L^{2} ({\mathbb {R}}^2 )\).

Keywords

Henstock-Kurzweil space Factorization theorem Convolution Distributional integral Tensor product 

Mathematics Subject Classification

26A39 46M05 43A32 42A85 46F12 

Notes

Acknowledgements

The authors express their sincere gratitude to Nancy Keranen for her excellent support. This work was partially supported by CONACyT-SNI, VIEP-BUAP, México.

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Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  • Oswaldo Flores-Medina
    • 1
    Email author
  • Juan H. Arredondo
    • 2
  • Juan A. Escamilla-Reyna
    • 1
  • Francisco J. Mendoza-Torres
    • 1
  1. 1.Benemérita Universidad Autónoma de Puebla, FCFMPueblaMexico
  2. 2.Department of MathematicsUniversidad Autónoma MetropolitanaIztapalapaMexico

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